實驗 7 ( 課本實驗 24) 氫原子光譜量度 與浦郎克常數

References 清華大學物理系普物實驗課本實驗 24 講義

目的 : 瞭解原子光譜 (atomic spectrum) 原理 : 氫原子 (H) 光譜 ( 質子 + 電子 ) 能階 : E n = -(m e e 4 /8  0 2 h 2 )/n 2 基態： n = 1, E 1 = eV 電子游離態： n = , E  = 0  E = E n1 – E n2 = hf = hc/ h = 6.63 x J-s c = 3.00 x 10 8 m/s Balmer series lines ( 巴爾麥系譜 線 ): n  2 Visible spectrum ( 可見光譜 ) H  : 3  2, = nm H  : 4  2, = nm H  : 5  2, = nm

實驗步驟 : 1. 利用氫氣管 (H 2 ) 高壓放電, 氫分子 (H 2 ) 分解成在激發態 (n  1) 之氫原 子觀察 (n  2) 之自發放射光譜 (spontaneous emission) ( 保護儀器  注意安全  2. 利用光譜儀 (spectrometer) 多狹縫光柵繞射 平行光入射光柵, 干涉條紋繞射條件 dsin  = m  (m = 0,  1,  2, …) m = 1: 三條譜線 (      ) m = 2: ~ 一條 (  ) 測量角度 (  ), 代入 d 及 m, 求波長 ( ) 鍵別率 :  = mN (N: 狹縫數 )

3. 與理論值比較 1/ = (m e e 4 /8  0 h 3 c)(1/ /n 2 ) (n  2) = (1/R H )(1/4 - 1/n 2 ) R H = 8.31 J/mol-K (Rydberg constant) 4. 利用實驗值波長代入 R H, 利用 m e = 9.11 x kg e = 1.60 x C  0 = 8.85 x F/m c = 3.00 x 10 8 m/s  求浦郎克常數 h, 與理論值 h = 6.63 x J-s 比較 5. 觀察其他氣體放電之可見光原子及分子光譜

DVD: 原子 (The Atom) (The Mechanical Universe…and Beyond/MU49) [Annenberg/CPB/ Bohr’s theory of hydrogen atom ( 波爾氫原子理論 ) Potential energy of electron bound to a proton U = -e 2 /4  0 r Total energy for circular orbit with centrifugal force mv 2 /r = e 2 /4  0 r 2 E = K + U = mv 2 /2 + U = - e 2 /8  0 r Frequency condition from spectral line ( 光譜頻率條件 )  E = E i – E f = hf if Quantized angular momentum L of the orbiting electron ( 電子軌道角 動量 ) L = mvr = n(h/2  ) n = 1, 2, 3, … Radii of allowed orbits r = n 2 h 2  0 /  me 2 = n 2 r B (r B = nm = 52.9 pm Bohr’s radius) Allowed energy states: E n = -(m e e 4 /8  0 2 h 2 )/n 2 = eV/n 2

Importance of the Hydrogen Atom The H-atom is the only atomic system that can be solved exactly. Much of what was learned about the H-atom, with its single electron, can be extended to such single-electron ions as He + and Li 2+. The H-atom proved to be an ideal system for performing precision tests of theory against experiment. Also for improving our understanding of atomic structure. The quantum numbers that are used to characterize the allowed states of hydrogen can also be used to investigate more complex atoms. This allows us to understand the periodic table. The basic ideas about atomic structure must be well understood before we attempt to deal with the complexities of molecular structures and the electronic structure of solids.

J. J. Thomson Atomic Model – Early Model (Newton’s Time) of the Atom J. J. Thomson established the charge to mass ratio for electrons. His model of the atom A volume of positive charge. Electrons embedded throughout the volume. The atom was a tiny, hard indestructible sphere. It was a particle model that ignored any internal structure. The model was a good basis for the kinetic theory of gases.

Rutherford’s Thin Foil Experiment Experiments done in A beam of positively charged alpha particles hit and are scattered from a thin foil target. Large deflections could not be explained by Thomson’s model. Rutherford Planetary model based on results of thin foil experiments Positive charge is concentrated in the center of the atom, called the nucleus. Electrons orbit the nucleus like planets orbit the sun

Difficulties with the Rutherford Model Atoms emit certain discrete characteristic frequencies of electromagnetic radiation. The Rutherford model is unable to explain this phenomena. Rutherford’s electrons are undergoing a centripetal acceleration. It should radiate electromagnetic waves of the same frequency. The radius should steadily decrease as this radiation is given off. The electron should eventually spiral into the nucleus.  But the fact doesn’t.

The Bohr Theory of Hydrogen -A Planetary Model of the Atom His model includes both classical and non-classical ideas. He applied Planck’s ideas of quantized energy levels to orbiting electrons. In this model, the electrons are generally confined to stable, nonradiating orbits called stationary states. Used Einstein’s concept of the photon to arrive at an expression for the frequency of radiation emitted when the atom makes a transition. In 1913 Bohr provided an explanation of atomic spectra that includes some features of the currently accepted theory.

The Bohr Model is probably familiar as the "planetary model" of the atom. for example, is used as a symbol for atomic energy (a bit of a misnomer, since the energy in "atomic energy" is actually the energy of the nucleus, rather than the entire atom). In the Bohr Model the neutrons and protons (symbolized by red and blue balls in the adjacent image) occupy a dense central region called the nucleus, and the electrons orbit the nucleus much like planets orbiting the Sun (but the orbits are not confined to a plane as is approximately true in the Solar System). The adjacent image is not to scale since in the realistic case the radius of the nucleus is about 100,000 times smaller than the radius of the entire atom, and as far as we can tell electrons are point particles without a physical extent.

This similarity between a planetary model and the Bohr Model of the atom ultimately arises because the attractive gravitational force in a solar system and  the attractive Coulomb (electrical) force between the positively charged nucleus and the negatively charged electrons in an atom are mathematically of the same form. The form is the same, but the intrinsic strength of the Coulomb interaction is much larger than that of the gravitational interaction; in addition, there are positive and negative electrical charges so the Coulomb interaction can be either attractive or repulsive, but gravitation is always attractive in our present Universe.

The Orbits Are Quantized -Quantized energy levels in hydrogen The basic feature of quantum mechanics that is incorporated in the Bohr Model. That is completely different from the analogous planetary model is that the energy of the particles in the Bohr atom is restricted to certain discrete values. One says that the energy is quantized.  This means that only certain orbit with certain radii are allowed;  orbits in between simply don't exist.

Quantized energy levels in hydrogen

Quantized Energy Levels in the hydrogen atom These energy levels are labeled by an integer n that is called a quantum number. The lowest energy state is generally termed the ground state. The states with successively more energy than the ground state are called the first excited state, the second excited state, and so on. Beyond an energy called the ionization potential the single electron of the hydrogen atom is no longer bound to the atom.  Then the energy levels form a continuum.  In the case of hydrogen, this continuum starts at 13.6 eV above the ground state ("eV" stands for "electron-Volt", a common unit of energy in atomic physics).

Atomic Excitation and De-excitation Atoms can make transitions between the orbits allowed by quantum mechanics by absorbing or emitting exactly the energy difference between the orbits. Excitation by absorption of light and de-excitation by emission of light

Atoms can make transitions between the orbits allowed by quantum mechanics by absorbing or emitting exactly the energy difference between the orbits. In each case the wavelength of the emitted or absorbed light is exactly such that the photon carries the energy difference between the two orbits. This energy may be calculated by dividing the product of the Planck constant and the speed of light hc by the wavelength of the light). Thus, an atom can absorb or emit only certain discrete wavelengths (or equivalently, frequencies or energies). Here is a Shockwave movie of atomic absorption and emission inShockwave movie Here is a Java applet illustrating atomic absorption and emission.Java applet

Separation of light by a prism according to wavelength Based on the Bohr atom, isolated atoms can absorb and emit packets of electromagnetic radiation having discrete energies dictated by the detailed atomic structure of the atoms.Bohr atom When the corresponding light is passed through a prism or spectrograph it is separated spatially according to wavelength.

Continuum, Emission & Absorption Spectra The corresponding spectrum may exhibit a continuum, or may have superposed on the continuum bright lines (an emission spectrum) or dark lines (an absorption spectrum), as illustrated in the following figure. Continuous Spectrum Emission Spectra Absorption spectra

Origin of Continuum, Emission & Absorption Spectra The emission spectra are produced by thin gases in which the atoms do not experience many collisions (because of the low density). The emission lines correspond to photons of discrete energies that are emitted when excited atomic states in the gas make transitions back to lower-lying levels. A continuum spectrum results when the gas pressures are higher. Generally, solids, liquids, or dense gases emit light at all wavelengths when heated. An absorption spectrum occurs when light passes through a cold, dilute gas and atoms in the gas absorb at characteristic frequencies; since the re-emitted light is unlikely to be emitted in the same direction as the absorbed photon, this gives rise to dark lines (absence of light) in the spectrum.

Sources of continuous, emission, and absorption spectra The emission spectra are produced by thin gases in which the atoms do not experience many collisions (because of the low density). The emission lines correspond to photons of discrete energies that are emitted when excited atomic states in the gas make transitions back to lower-lying levels. An absorption spectrum occurs when light passes through a cold, dilute gas and atoms in the gas absorb at characteristic frequencies; since the re-emitted light is unlikely to be emitted in the same direction as the absorbed photon, this gives rise to dark lines (absence of light) in the spectrum. Continuous Spectrum Emission Spectra Absorption spectra Hot Gas Cold Gas

Hydrogen emission series Hydrogen Emission & Absorption Series (visible light) (UV spectrum)

Hydrogen Emission & Absorption Series The spectrum of hydrogen is particularly important in astronomy because most of the Universe is made of hydrogen. Emission or absorption processes in hydrogen give rise to series, which are sequences of lines corresponding to atomic transitions, each ending or beginning with the same atomic state in hydrogen. The Balmer Series involves transitions starting (for absorption) or ending (for emission) with the first excited state of hydrogen. The Lyman Series involves transitions that start or end with the ground state of hydrogen. Because of the details of hydrogen's atomic structure, the Balmer Series is in the visible spectrum and the Lyman Series is in the the UV.

Because of the details of hydrogen's atomic structure, the Balmer Series is in the visible spectrum and the Lyman Series is in the the UV. The Balmer lines are designated by H with a greek subscript H i in order of decreasing wavelength. Thus the longest wavelength Balmer transition is designated H with a subscript alpha, H . the second longest H with a subscript beta, H , and so on, H , H  ….

Electron Transitions An electron transition in hydrogen between quantized energy levels with different quantum numbers n yields a photon by emission with quantum energy:quantized energy levelsemissionquantum energy This is often expressed in terms of the inverse wavelength or "wave number" as follows:

Quantized Energy States The electrons in free atoms can will be found in only certain discrete energy states. These sharp energy states are associated with the orbits or shells of electrons in an atom, e.g., a hydrogen atom. One of the implications of these quantized energy states is that only certain photon energies are allowed when electrons jump down from higher levels to lower levels, producing the hydrogen spectrum. The Bohr model successfully predicted the energies for the hydrogen atom, but had significant failures that were corrected by solving the Schrodinger equation for the hydrogen atom.photon energieshydrogen spectrumBohr modelsignificant failuresSchrodinger equation

Hydrogen Energy Levels

Basic Structure of the Hydrogen Energy Levels It can be calculated from the Schrodinger equation. The energy levels agree with the earlier Bohr model, and agree with experiment within a small fraction of an electron volt. If you look at the hydrogen energy levels at extremely high resolution, you do find evidence of some other small effects on the energy. The 2p level is split into a pair of lines by the spin-orbit effect.spin-orbit effect The 2s and 2p states are found to differ a small amount in what is called the Lamb shift.Lamb shift And even the 1s ground state is split by the interaction of electron spin and nuclear spin in what is called hyperfine structure.hyperfine structure

Balmer Line Series in Visible Spectrum Johann Jacob Balmer Analyzed the hydrogen spectrum and found that hydrogen emitted four bands of light within the visible spectrum: Balmer found that the data fit to the following equation: = wavelength (nm) R H = Rydberg's constant = x nm -1 n1 = the lower energy level n2 = the higher energy level Wavelength (nm) Color 656.2red 486.1blue 434.0blue-violet 410.1violet For example, to calculate the wavelength of light emitted when the electron in a hydrogen atom falls from the fourth energy level to the second energy level:

Each series is named after its discoverer: The Lyman series is the wavelengths in the ultra violet (UV) spectrum of the hydrogen atom, resulting from electrons dropping from higher energy levels into the n = 1 orbit. The Balmer series is the wavelengths in the visible light spectrum of the hydrogen atom, resulting from electrons falling from higher energy levels into the n = 2 orbit. The Paschen series is the wavelengths in the infrared spectrum of the hydrogen atom, resulting from electrons falling from higher energy levels into the n = 3 orbit. The Brackett series is the wavelengths in the infrared spectrum of the hydrogen atom, resulting from electrons falling from higher energy levels into the n = 4 orbit. The Pfund series is the wavelengths in the infrared spectrum of the hydrogen atom, resulting from electrons falling from higher energy levels into the n = 5 orbit.

Absorption Spectrum Joseph von Fraunhofer Studied the absorption spectrum of the light given off by the sun. Absorption Spectrum - The spectrum of dark lines against a light background that results from the absorption of selected frequencies of the electromagnetic radiation by an atom or molecule. The Balmer Series of Hydrogen consists of four visible lines.

The Balmer Series of Hydrogen (H) consists of four visible lines The helium (He) spectrum is somewhat more complex than that of hydrogen.

The neon (Ne) spectrum is dominated by red lines. The sodium (Na) spectrum consists of one very bright yellow line.

The mercury (Hg) spectrum